Spacetime is not just space and time

Our physical intuition usually separates space from time, ignoring the spacetime   character of the physical reality. In strongly curved spacetimes this may lead to confusion and paradoxes. I present here two examples: (1) in the non-static cosmological spacetimes with flat space sections, the cosmological expansion of space is a true physical effect. Contrary to what the intuition imagines, it cannot be explained as a motion of matter in a non-expanding flat space. (2) Contrary to intuition, for static spacetimes the mathematically simplest 3+13+1 split is not given by the direct projection as in the standard ADM scheme. The simplest split is defined by a counter intuitive “optical geometry” that redshifts both space and time by the same conformal factor.